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In geometric quantization, one of the important ingredients is an integrable distribution $D$ (let's say real) on some manifold $M$ (symplectic, but this is not important). The resulting object is $M/D$, the space of all maximal integral submanifolds of $M$ with respect to $D$.

Set-theoretically, given that through each point of $M$ there passes a single maximal such submanifold, there is a natural surjection $\pi : M \to M/D$. Geometric quantization is interested only in those $(M, D)$ such that there exist a smooth structure on $M/D$ making $\pi$ a submersion. My question is: is it possible to have multiple non-diffeomorphic such smooth structures on $M/D$, or is there a single one (and then $M/D$ could the be considered as "the" quotient space of $M$ by $D$)?

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    $\begingroup$ To clarify: are you asking whether the diffeomorphism class of the pair $(M, D)$ uniquely determines the smooth structure on $M/D$ (when it exists)? To add to Sebastian's answer below: If $M$ is fixed, and $D$ is allowed to vary, then $M/D$ can vary as well, e.g. there are certainly exist non-diffeomorphic manifolds that become diffeomorphic after product with $\mathbb R$. $\endgroup$
    – Igor Belegradek
    Commented Apr 23, 2016 at 18:28
  • $\begingroup$ @IgorBelegradek: $D$ is fixed. Indeed, I am asking whether there is a unique (up to diffeomorphisms) smooth structure on $M/D$ given a fixed pair $(M,D)$. $\endgroup$
    – Alex M.
    Commented Apr 23, 2016 at 18:35
  • $\begingroup$ @IgorBelegradek If $D$ is allowed to vary, many things can happen. For example, with $M=SU(2)$ and $D$ one-dimensional, one can get all Aloff-Wallach spaces and Eschenburg spaces. Some of them are homeomorphic but not diffeomorphic, some are not even homeomorphic, and so on. $\endgroup$
    – Sebastian Goette
    Commented Apr 24, 2016 at 21:45
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    $\begingroup$ @SebastianGoette: I know these examples. I was merely trying to clarify the question. $\endgroup$
    – Igor Belegradek
    Commented Apr 24, 2016 at 21:57

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Assuming there is a smooth structure on $M/D$ such that $\pi$ is a submersion. Then a function $f\colon M/D\to\mathbb R$ is smooth if and only if $f\circ\pi\colon M\to\mathbb R$ is smooth. This means, a homeomorphism $\psi\colon U\to V\subset\mathbb R^n$ for $U\subset M/D$ open (in the quotient topology) is a smooth chart for $M/D$ if and only if for all functions $g\colon V\to\mathbb R$, the composition $g\circ\psi\colon U\to\mathbb R$ is smooth if and only if $g$ is smooth. So the smooth structure is indeed unique if it exists.

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